GMM-based inference in the AR(1) panel data model for parameter values where local identi cation fails

Size: px
Start display at page:

Download "GMM-based inference in the AR(1) panel data model for parameter values where local identi cation fails"

Transcription

1 GMM-based inference in the AR() panel data model for parameter values where local identi cation fails Edith Madsen entre for Applied Microeconometrics (AM) Department of Economics, University of openhagen, Denmark April 9 PRELIMIARY Abstract We are interested in making inference on the AR parameter in an AR() panel data model when the time-series dimension is xed and the cross-section dimension is large. We consider a GMM estimator based on the second order moments of the observed variables. It turns out that when the AR parameter equals unity and certain restrictions apply to the other parameters in the model then local identi cation fails. We show that the parameters and in particular the AR parameter can still be estimated consistently but at a slower rate than usual. Also we derive the asymptotic distribution of the parameter estimators which turns out to be non-standard. The properties of this estimator of the AR parameter are very di erent from those of the widely used Arellano-Bond GMM estimator which is obtained by taking rst-di erences of the variables and then use lagged levels as instruments. It is well-known that this estimator performs poorly when the AR parameter is close to unity. The reason for this di erence between the two estimators is that with the Arellano-Bond moment conditions global identi cation fails for the speci c values of the parameters considered here whereas with the non-linear GMM estimator that uses all information from the second order moments only local identi cation fails. We illustrate the results in a simulation study. Keywords: AR() panel data model, GMM estimation, local identi cation failure, rate of convergence, non-standard limiting distribution

2 Introduction We are interested in making inference on the parameters in the simple AR() panel data model. This type of model is used to distinguish between two types of persistency often found in economic variables at the individual level such as incomes and wages. Within the framework of this model one source of persistency is attributed to the AR parameter which is common across all crosssection units and it describes how persistent unanticipated shocks to the variables of interest are. The other source of persistency is attributed to unobserved individual-speci c e ects and re ects di erences that remains constant over time for speci c cross-section units. Within this type of model the focus has mainly been on estimating the AR parameter whereas the individual-speci c e ects or the parameters describing speci c features of their distribution are treated as nuisance parameters. One of the most widely used estimators of the AR parameter is the Arellano-Bond GMM estimator which is obtained by taking rst-di erences of the variables in the equation in order to eliminate the individual-speci c e ects and then use lagged levels of the variable as instruments. This estimator has several advantages. First of all, it is based on moment conditions that are linear in the AR parameter and consequently it is straight forward to calculate. In addition, it only requires very basic assumptions about the di erent terms in the AR() panel data model. In particular, it does not require that the exact relationship between the initial values and the individual-speci c e ects is speci ed. However, it is also well-known that this estimator sometimes su ers from a weak instrument problem when the AR parameter is close to unity. The contribution of this paper is to get a better understanding of when there is a problem with identi cation (weak instruments in the linear case) and to show that the very basic assumptions implies moment conditions that do contain information about the AR parameter also when this is not the case for the linear Arellano-Bond moment conditions. Using all of the moment conditions gives a simple non-linear GMM estimator and we derive the asymptotic properties of this estimator. We will consider micro-panels where the cross-section dimension is large and the time-series dimension T is relatively small. This means that the asymptotic approximations to the statistics of interest will be based on letting! while T is xed. Under the same basic assumptions Ahn & Schmidt (995, 997) obtain the Arellano-Bond moments conditions and additional moment conditions that are quadratic in the AR parameter. These moment conditions do not depend on the remaining parameters in the model and they are obtained by transforming the original set of moment conditions. The approach in this paper is a somewhat di erent as we will use the full set of moment conditions and obtain estimators of all parameters used in the speci cation of the model. The ndings in this paper are based on a result from the statistical literature. Rotnitzky, ox, Bottai & Robins () consider a likelihood-based inference procedure when the standard assumptions about identi cation are not satis ed. The application to the GMM-based framework

3 is somewhat similar but there are also some di erences. We are able to derive the limiting distribution of the GMM estimator for the particular parameter values where local identi cation fails. It turns out that it is necessary to consider a Taylor expansion of the GMM objective function of higher order than usual in order to be able to explain its behavior. In particular, we nd that the GMM estimator of the AR parameter is =4 -consistent and has a limiting distribution which is a non-standard distribution. In a simulation study we show that this limiting distribution provides a good approximation to the actual distribution of the parameter estimators. Finally, the ndings in Stock & Wright () about the properties of GMM estimators under weak instruments are interesting in relation to the results of this paper. Stock & Wright () consider a general framework with moment conditions that are non-linear in the parameters but linear in the instruments under the assumption that the moment conditions provide only weak identi cation of one of the parameters. In this case they nd that the GMM estimator of the parameter that is only weakly identi ed is inconsistent whereas the GMM estimator of the remaining parameters is consistent at the usual rate but with a non-standard limiting distribution. In their paper the limiting distributions of the GMM estimators are expressed as being the distribution of the values at which a non-central chi-squared process is minimized. Here in this paper, the moment conditions provide identi cation of all parameters even though the assumption about rst order local identi cation is not satis ed which implies that the convergence rate is slower than usual. In addition we provide the exact limiting distribution. The model We consider the AR() panel data model with individual-speci c e ects y it = y it + i + " it for i = ; :::; and t = ; :::; T () For notational convenience it is assumed that y i is observed such that the number of observations over time equals T +. By repeated substitution in equation () we nd that y it = t y i +! Xt s i + s= The following assumptions are imposed: Assumption (y i ; i ; " i ; :::; " it ) is iid across i and has nite fourth order moments E (" it ) =, E " it = " and E (" is " it ) = for s 6= t and s; t = ; :::; T E ( i ) = and E i = E (y i ) = and E y i = E (" it i ) = and E (" it y i ) = for t = ; :::; T E (y i i ) = tx t s " is for t = ; ; ::: () s= 3

4 These assumptions are very general and in particular they do not require that the exact relationship between the initial values y i and the individual-speci c e ects i is speci ed. The assumptions imply that the observed variables y i = (y i ; :::; y it ) are iid across i with rst order moments equal to zero and second order moments that can be obtained by using the expression in (). In the next section this is used as the basis for the estimation of the parameters in the model. ote that we have assumed that the error terms " it have homoskedastic variances over time. This assumption could be relaxed in order to allow for heteroskedastic variances over time. Before we continue let us examine the assumption about mean stationarity. This assumption is often applied when deriving the asymptotic properties of parameter estimators in the simple AR() panel data model. It imposes a restriction on the relationship between the initial values and the individual-speci c e ects such that the covariance between the observed variable y it and the individual-speci c e ect i is the same for all t = ; ; :::; T. Mean stationarity can be formulated in the following two ways: ) y i = i + " i and i = ( ) i ) y i = i + " i for 6= where " i is iid across i with E (" i ) = and E " i = " for. Solving equation () recursively and using the assumption about mean stationarity we have ) y it = i + t " i + t " i + ::: + " it for all t = ; ; ; ::: (3) ) y it = i + t " i + t " i + ::: + " it for all t = ; ; ; ::: (4) This implies that (and therefore the term mean stationary ): ) E (y it j i ) = i for all t = ; ; ; ::: ) E (y it j i ) = i for all t = ; ; ; ::: As long as the AR parameter is considered as being xed the two ways of formulating mean stationarity are equivalent. However, when the value of approaches unity (usually formulated as being local-to-unity, i.e. = c= p ) then the two data generating processes are very di erent. In general under the formulation in ) we have = ( ) and = ( ) such that ( ) =. This means that = implies that = and =. It turns out that under the formulation in ) the correlation between y i and i is xed as! and zero when = whereas under the formulation in ) the correlation between y i and i tends to as! and in addition the variables y i ; :::; y it are dominated by the term i = ( to being linear dependent as!. ) and close From the expressions in (3)-(4) it is clear that the Arellano-Bond GMM estimator where lagged levels are used as instruments for the variables in rst-di erences su ers from a weak instrument 4

5 problem under mean stationarity and when is close to unity. Under the formulation in ) the rst-di erences y it can be written as y it = t t " i + ::: + ( ) " it + " it whereas the instruments y i ; :::; y it depend on i ; " i ; :::; " it. This means that there is a weak instrument problem when is close to unity. Under the formulation in ) this is even more serious as the instruments are dominated by the individual-speci c term i = ( ) which does not appear in the rst-di erences. In the existing literature it is common to consider the formulation in ). For example Blundell & Bond (998) and Blundell, Bond & Windmeijer () explain the weak instrument problem in the AR() panel data model within this framework from simulation studies and recently Hahn, Hausman & Kursteiner (7) have derived the asymptotic behavior of the Arellano-Bond GMM estimator under the assumption that is local-to-unity. In the present paper we derive the asymptotic properties of a speci c GMM estimator based on the restrictions in Assumption when the observed variables are generated by mean stationary AR() processes as formulated in ) with an AR parameter equal to unity. More speci cally, we consider the behavior of a GMM estimator when the true parameter values are = ; = and =. It is important to emphasize that only the restrictions in Assumption and not the additional restrictions implied by mean stationarity are used in the estimation of the parameters. 3 Estimation For the moment, we assume that we observe y i ; y i ; y i for all i (3 observations over time). The model contains the following 5 parameters: ; ; ; ; " The 5 vector of parameters is de ned as = ; ; ; ; ". We let denote the true value of the parameter. In the following we are interested in estimation of the parameter when its true value equals with = ; ; ; ; " where and " are the true (unknown) values of the parameters and ". By using the expression in () we have that y i = y i + i + " i y i = y i + ( + ) i + " i + " i 5

6 Under Assumption the model yields the following 6 moment conditions: E yi = E (y i y i ) = + E (y i y i ) = + ( + ) E yi = " E (y i y i ) = 3 + ( + ) + ( + ) + " E yi = 4 + ( + ) + ( + ) + + " Using vector notation this can be written as E [g (y i ; )] = (5) where the 6 vector g (y i ; ) corresponding to the expressions above is de ned in Appendix A. The moment conditions are based on second order moments of the observed variable y i = (y i ; y i ; y i ). We see that for a xed value of then the expressions on the right hand side in the equations above are linear in the remaining parameters ; ; ; ". ote that we do not use the additional moment conditions implied by mean stationarity here. The variance of moment function g (y i ; ) evaluated at the true parameter value = is given by Var [g (y i ; )] = E g (y i ; ) g (y i ; ) = (6) where is positive de nite a 6 6 matrix that depends on the fourth order moments of the observed variables (y i ; y i ; y i ). Ahn & Schmidt (995, 997) show that under Assumption and when there are 3 observations over time then the following two moment conditions hold E (y i (y i y i )) = (7) E (y i y i ) (y i y i ) = (8) The rst is the linear Arellano-Bond moment condition and the other is quadratic in the AR parameter. ote that they do not depend on the variance parameters ; ; ; ". These two moment conditions consist of a subset of the full set of moment conditions in (5) and therefore there might be a loss of information by only using these two moment conditions. repon, Kramarz & Trognon (997) show that when the usual rst order asymptotic approximations apply then the GMM estimator of the AR parameter based on the moment conditions in (5) and the GMM estimator based on the two moments above are asymptotically equivalent. This might not be the case here in this paper since the rst order asymptotic approximations are not valid. Therefore we will consider the full set of moment conditions in (5). 6

7 3. Identi cation The condition for global identi cation is The condition for rst order local identi cation rank (y i; ) E [g (y i ; )] = if and only if = (9) = 5 () It turns out that rst order local identi cation fails at rank (y i; ) = (y i; ) = 4 () In Appendix A it is shown that the derivatives with respect to ; ; ; " are linear dependent. This does not imply that these parameters are not identi ed when = but it does mean that the behavior of the GMM estimator can not be explained by the usual rst-order asymptotic approximations. When () holds then there is a reparametrization = () such that = and ~ = ( ; ::; 5 ) and where it holds that the expected value of the rst order derivative of g (y i ; ) with respect to equals zero at = ( ) and the 6 4 matrix of the expected value of the rst order derivatives of g (y i ; ) with respect to ~ has full rank equal to 4 at = ( ). This means that in a neighborhood around the moment conditions can be written ~ E [g (y i ; )] = ~ (y i; ) g + E (y i ; ) =! ( ) () The right hand side in the expression above equals zero if and only if = when the following condition @g (y i ; ) 6= and it is not a linear combination of ~ (y i; ) In particular, this means that rst order local identi cation is satis ed for the parameters ( ) = ( ) and ( ~ ~ ). It turns out that (3) is satis ed in the case we consider in this paper. An implication of this is that when the true value of equals it is possible to estimate the parameters ( ) and ( ~ ~ ) consistently at the usual p -rate. This is shown in the next section where we also show that it is possible to estimate the parameter of interest consistently at a rate of =4 when its true value equals. Let us brie y have a look at the moment conditions underlying the Arellano-Bond GMM estimator. They are given by E [(y i y i ) y i ] = E yi (3) ( ) E (y i y i ) + E (y i y i ) (4) They only depend on and are linear in this parameter. With moment conditions that are linear in the parameters it holds that the concepts of rst order local and global identi cation are equivalent. 7

8 It turns out that identi cation fails when = and = and in particular mean-stationarity and an AR parameter equal to unity lead to lack of identi cation. In general identi cation fails when E y i = E (yi y i ) = E (y i y i ). ote that the Arellano-Bond moment conditions in (4) can be written as linear combinations of the moment conditions in (5). In particular, they are a subset of the moment conditions in (5) meaning that they contain less information about the AR parameter. 3. The GMM estimator We consider the GMM estimator ^ that minimizes a quadratic form in the sample moments P i= g (y i; ), that is where The weight matrix W i= ^ = arg min Q () (5) " # " # X X Q () = g (y i ; ) W g (y i ; ) i= (6) is positive semi-de nite and may depend on the data but converges in probability to a positive de nite matrix W as!. In the following we will use the identity matrix as a weight matrix. This is done for simplicity and a di erent weight matrix might be used in the future. Also it is not clear if the weight matrix that is found to be optimal under the standard assumption about local identi cation is optimal when this assumption fails. Since we are rst of all interested in the AR parameter and since the moment conditions in (5) for xed are linear in the other parameters we will consider the concentrated objective function. For a xed value of we let ^ () denote the GMM estimator of = ; ; ; ", that is ^ () = arg min Q (; ) (7) In Appendix A we show that for any value of the minimization problem above always have a unique solution and a closed-form expression for ^ () is easily obtained. In addition ^ ( ) is a p -consistent estimator of if the true value of equals. This results from the matrix in () having reduced rank equal to 4 (the number of parameters minus ) and it is crucial for the results in the paper to hold. The concentrated objective function is de ned as Q () = Q ; ^ () = g () g () (8) where g () = W = P i= g y i ; ^ () is the sample moments evaluated at ; ^ (). ote here that ^ () depends on the weight matrix W but for simplicity this is suppressed in the notation. The GMM estimator of can then be expressed as ^ = arg min Q () (9) 8

9 When we use the usual optimal weight matrix, i.e W =, then asymptotically times the concentrated objective function as a function of is a non-central -distribution with degrees of freedom (this results from starting with 6 moment conditions and concentrating out 4 parameters). The non-centrality parameter depends on and is given by F () I 6 F () F () F () F () F () where F () = = S (). In particular, since the non-centrality parameter equals zero at = we have that asymptotically Q () is a -distribution with degrees of freedom. It is also clear that the non-centrality parameter is increasing in the number of cross-section observations. Altogether we might consider Q () as a non-central chi-squared process indexed by. It is then possible to simulate this process for di erent values of and obtain the value of where the minimum is obtained. The distribution of these values of would then re ect the asymptotic distribution of the GMM estimator ^. This approach is used in Stock & Wright () within a weak instrument framework such that the non-centrality parameter is considered as being constant as tends to in nity. However, here we will use an approximation of the concentrated objective function in a neighborhood around unity in order to explain the behavior of the GMM estimator ^. 3.3 Asymptotic approximations The derivative of order k of the concentrated objective function is de ned as Q (k) () = dk Q () () dk Expressions can be found in Appendix A. A Taylor expansion of order k of the concentrated objective function Q () around the true value gives Q () = Q ( ) + kx j= Q (j) ( ) ( ) j =j! + L k () () where L k () = Q (k+) () ( ) k+ = (k + )! for some with j j < j j. When the "standard regularity conditions" (including rst order local identi cation) are satis ed then the following results concerning derivatives of the concentrated objective function hold p () Q ( ) = g () ( ) p g ( )! w ; 4G ~ G as! () Q () ( ) as = g () ( ) g () ( )! P G G > as! (3) Q (3) ( ) = O P () (4) where we have used that p g ( )! w ; ~ ; g () ( )! P G 6= and g (k) ( )! P G k as!. For p ( ) bounded then the expansion in () together with the results above imply that the centered (equal to zero at = ) and appropriate scaled concentrated objective 9

10 function can be approximated by a second order polynomial in p ( ) in a neighborhood around. We have the following (Q () Q ( )) as = p np o np Q () ( ) ( ) + Q () ( ) =! ( )o (5) When considered as a function of p ( unique minimum which is asymptotically equivalent to p (^ ) the right hand side in the expression above has a of which minimizes the left hand side in the expression above. This gives p (^ ) as = p Q () ( ) =Q () ( ) w! ) since ^ is de ned as the value ; (G G ) G ~ G (G G ) as! (6) It means that under the "standard regularity conditions" (including rst order local identi cation) the GMM estimator ^ is a p -consistent estimator of and it is asymptotically normally distributed. In the case considered here in this paper when local identi cation fails for = we need a Taylor expansion of higher order in order to derive asymptotic properties of the GMM estimator. We have the following as! (for details see Appendix A) 3=4 Q () () = =4 g () p () g ()! P as! (7) p () as Q () = g () p () g ()! w ; 4G ~ G as! (8) =4 Q (3) () P! as! (9) Q (4) as () = 6g () () g () ()! P 6G G > as! (3) Q (5) as () = g () () g (3) ()! P G G 3 as! (3) where we have used that p g ()! w ; ~ ; =4 g () ()! P, g () ()! P G 6= and g (k) () P! G k as!. For =4 ( ) bounded then the expansion in () together with the results above imply that the centered and scaled concentrated objective function can be approximated by a fourth order polynomial in =4 ( have the following ) in a neighborhood around unity. We = (Q () Q ()) n o pq o =4 () ( ) () =! + Q(4) n () =4! =4 ( ) + R () (3) where R () =! () 3=4 Q () () + =4 Q (3) () =3!! () + =4 Q (5) () =5!! ()4 (33) n o! () = =4 ( ) (34) It holds that R () P! as!. otice that the limiting distributions of the coe cients in the polynomial on the right hand side in equation (3) above are of the same form as in the

11 second order polynomial obtained when the assumption about local identi cation is satis ed. We have the following (Q () n o Q ()) as n o = =4 ( ) Z ~ + =4G G =4 ( ) where the right and left hand side in the expression above are asymptotically equivalent and ~Z ; G ~ G. In the case where we have a positive value of Z ~ (this will happen with probability /) then the right hand side in the expression above has a global minimum at =. In the case where we have a negative value of ~ Z (this will happen with probability /) then the expression on the right hand side will have a local maximum at = and two global minima attained at =4 ( ) = q ~Z= (=G G ). This means that ^ = with probability / and ^ = =4 q ~Z= (=G G ) with probability /. In order to determine which of these values gives the minimum of the objective function we will need to consider the deviation of the scaled objective function and the expression on the right hand side in the expression above evaluated at the two values. Using equation (33) we write R () =! () T () where T () = 3=4 Q () () + =4 Q (3) () =3!! () + =4 Q (5) () =5!! ()4 (36) q q We see that T (^ ) = T (^ ) for ^ = ~Z= (=G G )= =4 and ^ = + ~Z= (=G G )= =4. This means that the sign of T (^) will determine at which of the two values the minimum is attained. If the sign of T () is negative (positive) the minimum is attained at ^ (^ ). This has to be conditional on getting a negative draw of p Q () as () =! = Z. ~ So in order to determine where the minimum is attained we have to consider the conditional distribution of =4 T (^) given p () Q () =! and determine the probability that T (^) < conditional on p Q () () <. The di erence between the framework considered here in this paper and the one in Rotnitzky, ox, Bottai & Robins () is that the expected value of the rst h order i derivative of the concentrated moment function at equal to unity equals zero, i.e. E g () () =, while Rotnitzky, ox, Bottai & Robins () assume that the rst order derivative of their objective function (the log-likelihood function) at this point is equal to zero with probability. This means that in our framework p g () () = O P () such that expressions in =4 T (^) involving this term will appear and that complicate things. In fact it turns out that if the usual optimal weight matrix W = is used and if it was the case that g () () = then we would have that =4 T (^) is asymptotically normal with mean zero and independent of p Q () () =! such that the minimum is attained at ^ and ^ with equal probability. (35) The limiting distribution of the GMM estimator of the AR parameter ^ is given in the proposition below. Proposition Under Assumption and when the true value of equals then the following holds: =4 (^ ) w! (Z>) + (Z<) ( ) B p Z as! (37)

12 where B Ber (:5) and Z (; ) with = 4 (G G ) G ~ G (G G ). In addition B and Z are independent of each other (as explained above this is not shown yet). The result shows that ^ is a =4 -consistent estimator of the AR parameter and that its asymptotic distribution is a mixture with 5% of the probability mass at the true value of unity and 5% each at unity plus/minus the square root of a half-normal distribution. This gives a distribution which is symmetric around the true value of unity with 3 modes of which one is at unity and where the two other modes will become closer to unity as the sample size increases. Also note that the result implies that the parameter ( ) can be estimated consistently at the usual p -rate and that the limiting distribution of p (^ ) is a mixture with 5% of the probability mass at the true value which equals and 5% distributed according to the positive half-normal of Z. This result is similar to what is found for the behavior of parameter estimators when the true value of the parameter is on the boundary of the parameter space, see Geyer (994) and Andrews (999). The limiting distribution of the GMM estimator of the remaining parameters follows by the results in the proposition below. Proposition Under Assumption and when the true value of equals then the following holds as! : p ^ (^) H (^ ) H (^ ) w! Z (38) p ^ (^) H (^ ) H (^ ) =4 (^ ) w! (Z j Z ) (39) where Z Z ; (4) with = 4 (G G ) G ~ G (G G ) H = 6 4 = M () W = W = M () = (G G ) G ~ W = M () M () = F () F () F () " " H = M () F () () H F () () The result in (38) shows which possibly non-linear transformations of the parameter vector that can be estimated consistently at the usual p -rate when the true value of equals

13 . The result also gives the marginal asymptotic distributions of the GMM estimator of the parameter since it implies that ^ =4 (^) as = H (^ ). In particular, this means as = whereas the asymptotic distributions of the remaining parameters are that =4 ^ similar to the one obtained for the GMM estimator of the AR parameter ^, see Proposition. In particular, they can be estimated consistently at the rate =4. The result also shows that when using the usual optimal weight matrix W = then the estimator of the transformation of the parameters that can be estimated p -consistently is asymptotically independent of the estimator ^. The result in (39) shows the conditional distribution of p ^ (^) given the value of =4 (^ ). In general we nd that conditional on Z (see Proposition ) then ^ (^) is asymptotically normal with mean + H (^ ) + H (^ ) Z and variance =. Since =4 (^ ) = p Z for Z < and =4 (^ ) = for Z > this means that conditional on (^ ) then with 5% probability ^ (^) is asymptotically normal with mean + E (Z j Z < ) and with 5% probability ^ (^) is asymptotically normal with mean + H (^ ) + H + (^ ). When the usual optimal weight matrix is used, that is W =, then conditional on (^ ) then with 5% probability ^ (^) is asymptotically normal with mean and with 5% probability ^ (^) is asymptotically normal with mean + H (^ ) + H (^ ). 4 A simulation study The results derived in the previous section are now illustrated in a simulation study. We consider the AR() panel data model with parameter values where local identi cation fails, i.e. = ; = ; = ; = ; " =. We consider 5 replications of the model: y it = y it + " it for i = ; :::; and t = ; y i iid ; across i " it iid ; " across i; t We have that = and = 5. In practice the minimum of the concentrated objective function Q () is found by a grid search in the interval [:; :8]. The reason for this is that the function Q () in approximately 5% of the replications have two local minima of which one is the global minimum. We have used the weight matrix W = where = Var [g (y i ; )]. Overall the outcome of this simulation experiment shows that the asymptotic results derived in the previous section explains the actual behavior of the GMM estimator of the AR parameter well (Figure -4). In addition the outcome shows that the joint distribution of the estimators of is also very di erent from what we usually nd when they are jointly normally distributed. (Figure 5-8). This follows by the result in Proposition which also explains the strong quadratic relationship (the green plots) between the estimators of for instance and. 3

14 Figure : Empirical distribution of ^ for = Figure : Empirical distribution of ^ when p Q () () < for = 4

15 Figure 3: Empirical distribution of ^ for = 5 Figure 4: Empirical distribution of ^ when p Q () () < for = 5 5

16 Figure 5: Plot of (^; ^ ) for = 5. The red lines are the true parameter values Figure 6: Plot of (^; ^ ) for = 5. The red lines are the true parameter values 6

17 Figure 7: Plot of (^; ^ ) for = 5. The red lines are the true parameter values Figure 8: Plot of (^; ^ " ) for = 5. The red lines are the true parameter values 7

18 Appendix A In this appendix the notation "X as = Y " means that X and Y are asymptotically equivalent as! such that X Y P! as!. The moment conditions: The 6 vector g (y i ; ) consists of the following elements g (y i ; ) = yi g (y i ; ) = y i y i + g 3 (y i ; ) = y i y i + ( + ) g 4 (y i ; ) = yi " g 5 (y i ; ) = y i y i 3 + ( + ) + ( + ) + " g 6 (y i ; ) = yi 4 + ( + ) + ( + ) + + " ote that they can be written as g (y i ; ) = vech (y i y i) S () (4) where = ; ; ; " and the 6 4 matrix S () is de ned as We have that S () = 6 4 ( + ) 3 ( + ) ( + ) 4 ( + ) ( + (y i; ) = h S () ().. S () where S () () denotes the rst order derivative of S () and is given by 3 S () () = ( + ) i (4) (43) (44) We see that S () () = S () q with q = ; " ; ; " such that the matrix in (43) has reduced rank equal to 4 when = = ; ; ; ; ". 8

19 We have E [g (y i ; )] = (45) E g (y i ; ) g (y i ; ) = (positive de nite) (46) such that p X i= X i= g (y i ; ) w! (; ) as! (47) g (y i ; )! X g (y i ; ) i= g (y i ; ) X P g (y i ; )!! as! (48) i= The estimator ^ (): Using the expression in (43) we have that () X g (y i ; )# W S () (49) i= The FO for the optimization problem in (7) give " X vech (y i y i) S () ^ ()# W S () = (5) i= The FO are necessary and su cient since S () W S () as = S () W S () > for all values of since W is positive de nite and S () has full rank equal to 4 for all values of. This implies that ^ () = S () W S () S () W X vech (y i yi) (5) i= The concentrated objective function Q () and its derivatives: We de ne the concentrated sample moment function g () = W = X i= g y i ; ; ^ () = W = The concentrated objective function can be written as X vech (y i yi) i= S () ^ ()! (5) Q () = g () g () (53) 9

20 Using this we obtain the following expressions for derivatives of Q Q () () = g () g () () (54) Q () () = g () g () () + g() () g () () (55) Q (3) () = g () g (3) () + 6g() () g () () (56) Q (4) () = g () g (4) () + 8g() () g (3) () + 6g() () g () () (57) Q (5) () = g () g (5) () + g() () g (4) () + g() () g (3) () (58) Inserting the expression for ^ () in (5) in equation (5) we have g () = I 6 F () F () F () F () W = X vech (yiy i ) (59) where F () = W = S (). ote that I 6 F () F () F () F () W = S () = such that when the true value of equals then the concentrated moment function evaluated at the true value can be written as g ( ) = I 6 F ( ) F ( ) F ( ) F ( ) W = i= X g (y i ; ) (6) The Lemma below gives the limiting distributions of p g () and p ^ () when the true value of equal. Lemma 3 When = the following holds as! where M () W = p (I 6 P ()) W = For W = we have that X i= p g (y i ; ) w! X i= g (y i ; ) w! F () = W = F () = W = S () i= ; M () W = W = M () (6) ; (I 6 P ()) W = W = (I 6 P ()) (6) (y i ; ) S () = M () = F () F () F () M () = F () F () F () P () = F () M () P () = F () M () (I 6 P ()) W = W = M () =

21 such that the two linear transformations of P i= g (y i; ) are orthogonal. The probability limit of ^ () when the true parameter value equals is given by where plim ^ () = M () S () (63)! M () = S () W S () S () W (64) In addition we have that p ^ () M () S () w! ; M () M () as! (65) This means that ^ () is a p -consistent estimator of since M () S () = I 4 or more generally that when evaluated at the true value of the estimator ^ is a p -consistent estimator of the true value of. Asymptotic results: For the rst order derivative of Q () we have that 3=4 Q () () = = g () W =4 g () ()! P as! (66) where we have used that = g ()! w ; ~ and that =4 g () ()! P as!. For the second order derivative of Q () we have that = Q () () =! = = g () g () () + =4 g () () =4 g () () as = = g () g () ()! w ; G ~ G as! (67) where we have used that = g ()! w ; ~, =4 g () ()! P and that g () ()! P G as!. For the third order derivative of Q () we have that =4 Q (3) ()!=3! = =3 =4 g () g (3) () + =4 g () () g () () P! as! (68) where we have used that =4 g ()! P and =4 g () ()! P as! and also that g () () and g (3) () both converge in probability as!. For the fourth order derivative of Q () we have that Q (4) () =4! = =g () g (4) () + =3g() () g (3) () + =4g() () g () () as = =4g () () g () ()! P =4G G as! (69)

22 where we have used that g ()! P and g () ()! P as! and also that g (3) () and g(4) () both converge in probability as!. For the fth order derivative of Q () we have that =4 Q (5) () =5! = =5! =4 g () g (5) () + =5! =4 g () () g (4) () + =5! =4 g () () g (3) () P! as! (7) where we use arguments similar to the ones above. This gives the following result (Q (^) Q ()) as = 3=4 Q () () =4 (^ ) + = Q () n () =4 (^ )o =! + =4 Q (3) () n =4 (^ ) o 3 4 (4) =3! + Q n () =4 (^ )o =4! 5 + =4 Q (5) n () =4 (^ )o =5! o as 4 = = g () g () n () =4 () (^ ) + =4g () g () n () =4 (^ )o (7) Derivatives of the concentrated moment function: g k () = dk g () (7) dk The rst order derivative g () = = W = S () () ^ () () S () ^ () (73) The second order derivative g () = W = S () () ^ () S () () () () ^ () S () ^ () (74) The third order derivative g (3) = W = S (3) () ^ () 3S () () () ^ 3S () () (3) () ^ () S () ^ () The fourth order derivative g (4) = W = S (4) () ^ () 4S (3) () () ^ 6S () () () ^ 3S () (3) () ^ S () (4) ^ (75) (76)

23 Limiting distribution of ^ (^): We use the following de nitions The rst order derivative of ^ (): ^() () = F () F () () F () W = F () = W = S () (77) F (k) = () = W S(k) () for k = ; ; 3; 4 (78) X vech (y i yi) i= F () F () h i F () () F () + F () S () F () () F () F () W = = F () F () () W = F () Using this we have that when = X vech (y i yi) i= h F () () F () + F () F () X vech (y i yi) i= () i ^ ()! =4^() () as = =4 F () F () F () () F () h i F () () F () + F () F () () = =4 F () F () F () F () () = =4 q (79) where the second line follows by using that p P i= (vech (y iy i ) S () ) w! (; ) as! and the last line follows from using that F () () = W = S () () = W = S () q = F () q. with q = ; " ; ; " The second order derivative of ^ (): ^() () = F () F () () (F () W = X vech (y i y i) i= F () F () h i F () () F () + F () () F () () + F () F () () ^ () F () F () h F () () F () + F () F () () i ^() () 3

24 Using this we have that when = as =4^() () = =4 F () F () (F () () F () h i + F () () F () + F () F () () q) = F () F () h F () () F () () + F () F ()i () =4 + F () F () h i F () () F () + F () F () () =4 q = =4 F () F () F () F () () q F () () h i F () () F () + F () () F () () + F () F () () P where the second line follows by using that p i= (vech (y iyi ) S () )! w (; ) as! and that ^ =4 () P! and =4 ^() P! () + q as! and the third line follows from using that S () q = S () (). (8) The limiting distribution of ^ (^) when the true value equals unity and the true value of equals = ; ; ; " p ^ (^) ^() () (^ ) ^() () (^ ) =! = p ^ (^) ^ () ^() () (^ ) ^() () (^ ) =! + ^ () as = w! p ^ () ; F () F () F () W = W = F () F () F () as! where we have used the result in (65) and that p ^ (^) ^ () ^() () (^ ) ^() () (^ ) =! = =4^(3) n 3 () =3! =4 (^ )o = op () This gives the following asymptotic distribution of ^ (^) =4 ^ (^) as () = ^ () =4 (^ ) + =4^() n o () =! =4 (^ ) as = q =4 (^ ) 4

25 Derivatives of S (): S () () = S () () = S (3) () = S (4) () = ( + ) (8) (8) (83) (84) 5

26 References Ahn, S.. and P. Smith, 995, E cient estimation of models for dynamic panel data, Journal of Econometrics 68, 5-7. Ahn, S.. and P. Smith, 997, E cient estimation of models for dynamic panel data: Alternative assumptions and simpli ed estimation, Journal of Econometrics 76, Andrews, D.W.K., 999, Estimation when a parameter is on a boundary, Econometrica 67, Blundell, R. and S. Bond, 998, Initial conditions and moment restrictions in a dynamic panel data models, Journal of Econometrics 87, Blundell, R., S. Bond and F. Windmeijer,, Estimation in dynamic panel data models: Improving on the performance of the standard GMM estimator, in: Baltagi, B.H. (Ed), Advances in Econometrics vol. 5. repon, B., F. Kramarz and A. Trognon, 997, Parameters of interest, nuisance parameters and orthogonality conditions: An application to autoregressive error component models, Journal of Econometrics 8, Geyer,.J., 994, On the asymptotics of constrained M-estimation, The Annals of Statistics, Hahn, J., J. Hausman and G. Kursteiner, 7, Long di erence instrumental variables estimation for dynamic panel models with xed e ects, Journal of Econometrics 4, Madsen, E., 9, GMM estimators and unit root test in the AR() panel data model (submitted for publication). Rotnitzky, A., D.R. ox, M. Bottai and J. Robins,, Likelihood-based inference with singular information matrix, Bernoulli 6(), Stock, J.H. and J.H. Wright,, GMM with weak identi cation, Econometrica 68,

Chapter 1. GMM: Basic Concepts

Chapter 1. GMM: Basic Concepts Chapter 1. GMM: Basic Concepts Contents 1 Motivating Examples 1 1.1 Instrumental variable estimator....................... 1 1.2 Estimating parameters in monetary policy rules.............. 2 1.3 Estimating

More information

Chapter 2. Dynamic panel data models

Chapter 2. Dynamic panel data models Chapter 2. Dynamic panel data models School of Economics and Management - University of Geneva Christophe Hurlin, Université of Orléans University of Orléans April 2018 C. Hurlin (University of Orléans)

More information

GMM based inference for panel data models

GMM based inference for panel data models GMM based inference for panel data models Maurice J.G. Bun and Frank Kleibergen y this version: 24 February 2010 JEL-code: C13; C23 Keywords: dynamic panel data model, Generalized Method of Moments, weak

More information

On GMM Estimation and Inference with Bootstrap Bias-Correction in Linear Panel Data Models

On GMM Estimation and Inference with Bootstrap Bias-Correction in Linear Panel Data Models On GMM Estimation and Inference with Bootstrap Bias-Correction in Linear Panel Data Models Takashi Yamagata y Department of Economics and Related Studies, University of York, Heslington, York, UK January

More information

Inference about Clustering and Parametric. Assumptions in Covariance Matrix Estimation

Inference about Clustering and Parametric. Assumptions in Covariance Matrix Estimation Inference about Clustering and Parametric Assumptions in Covariance Matrix Estimation Mikko Packalen y Tony Wirjanto z 26 November 2010 Abstract Selecting an estimator for the variance covariance matrix

More information

Time Series Models and Inference. James L. Powell Department of Economics University of California, Berkeley

Time Series Models and Inference. James L. Powell Department of Economics University of California, Berkeley Time Series Models and Inference James L. Powell Department of Economics University of California, Berkeley Overview In contrast to the classical linear regression model, in which the components of the

More information

13 Endogeneity and Nonparametric IV

13 Endogeneity and Nonparametric IV 13 Endogeneity and Nonparametric IV 13.1 Nonparametric Endogeneity A nonparametric IV equation is Y i = g (X i ) + e i (1) E (e i j i ) = 0 In this model, some elements of X i are potentially endogenous,

More information

On the optimal weighting matrix for the GMM system estimator in dynamic panel data models

On the optimal weighting matrix for the GMM system estimator in dynamic panel data models Discussion Paper: 007/08 On the optimal weighting matrix for the GMM system estimator in dynamic panel data models Jan F Kiviet wwwfeeuvanl/ke/uva-econometrics Amsterdam School of Economics Department

More information

Chapter 2. GMM: Estimating Rational Expectations Models

Chapter 2. GMM: Estimating Rational Expectations Models Chapter 2. GMM: Estimating Rational Expectations Models Contents 1 Introduction 1 2 Step 1: Solve the model and obtain Euler equations 2 3 Step 2: Formulate moment restrictions 3 4 Step 3: Estimation and

More information

Simple Estimators for Semiparametric Multinomial Choice Models

Simple Estimators for Semiparametric Multinomial Choice Models Simple Estimators for Semiparametric Multinomial Choice Models James L. Powell and Paul A. Ruud University of California, Berkeley March 2008 Preliminary and Incomplete Comments Welcome Abstract This paper

More information

Simple Estimators for Monotone Index Models

Simple Estimators for Monotone Index Models Simple Estimators for Monotone Index Models Hyungtaik Ahn Dongguk University, Hidehiko Ichimura University College London, James L. Powell University of California, Berkeley (powell@econ.berkeley.edu)

More information

Panel Data. March 2, () Applied Economoetrics: Topic 6 March 2, / 43

Panel Data. March 2, () Applied Economoetrics: Topic 6 March 2, / 43 Panel Data March 2, 212 () Applied Economoetrics: Topic March 2, 212 1 / 43 Overview Many economic applications involve panel data. Panel data has both cross-sectional and time series aspects. Regression

More information

On Standard Inference for GMM with Seeming Local Identi cation Failure

On Standard Inference for GMM with Seeming Local Identi cation Failure On Standard Inference for GMM with Seeming Local Identi cation Failure Ji Hyung Lee y Zhipeng Liao z First Version: April 4; This Version: December, 4 Abstract This paper studies the GMM estimation and

More information

ECONOMETRICS FIELD EXAM Michigan State University May 9, 2008

ECONOMETRICS FIELD EXAM Michigan State University May 9, 2008 ECONOMETRICS FIELD EXAM Michigan State University May 9, 2008 Instructions: Answer all four (4) questions. Point totals for each question are given in parenthesis; there are 00 points possible. Within

More information

Strength and weakness of instruments in IV and GMM estimation of dynamic panel data models

Strength and weakness of instruments in IV and GMM estimation of dynamic panel data models Strength and weakness of instruments in IV and GMM estimation of dynamic panel data models Jan F. Kiviet (University of Amsterdam & Tinbergen Institute) preliminary version: January 2009 JEL-code: C3;

More information

Estimating the Number of Common Factors in Serially Dependent Approximate Factor Models

Estimating the Number of Common Factors in Serially Dependent Approximate Factor Models Estimating the Number of Common Factors in Serially Dependent Approximate Factor Models Ryan Greenaway-McGrevy y Bureau of Economic Analysis Chirok Han Korea University February 7, 202 Donggyu Sul University

More information

Parametric Inference on Strong Dependence

Parametric Inference on Strong Dependence Parametric Inference on Strong Dependence Peter M. Robinson London School of Economics Based on joint work with Javier Hualde: Javier Hualde and Peter M. Robinson: Gaussian Pseudo-Maximum Likelihood Estimation

More information

A Course on Advanced Econometrics

A Course on Advanced Econometrics A Course on Advanced Econometrics Yongmiao Hong The Ernest S. Liu Professor of Economics & International Studies Cornell University Course Introduction: Modern economies are full of uncertainties and risk.

More information

A New Approach to Robust Inference in Cointegration

A New Approach to Robust Inference in Cointegration A New Approach to Robust Inference in Cointegration Sainan Jin Guanghua School of Management, Peking University Peter C. B. Phillips Cowles Foundation, Yale University, University of Auckland & University

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 6 Jakub Mućk Econometrics of Panel Data Meeting # 6 1 / 36 Outline 1 The First-Difference (FD) estimator 2 Dynamic panel data models 3 The Anderson and Hsiao

More information

Notes on Time Series Modeling

Notes on Time Series Modeling Notes on Time Series Modeling Garey Ramey University of California, San Diego January 17 1 Stationary processes De nition A stochastic process is any set of random variables y t indexed by t T : fy t g

More information

Testing for Regime Switching: A Comment

Testing for Regime Switching: A Comment Testing for Regime Switching: A Comment Andrew V. Carter Department of Statistics University of California, Santa Barbara Douglas G. Steigerwald Department of Economics University of California Santa Barbara

More information

Judging contending estimators by simulation: tournaments in dynamic panel data models

Judging contending estimators by simulation: tournaments in dynamic panel data models Discussion Paper: 005/0 Judging contending estimators by simulation: tournaments in dynamic panel data models Jan F. Kiviet Appeared as Chapter (pp.8-38) in: The Refinement of Econometric Estimation and

More information

Robust Standard Errors in Transformed Likelihood Estimation of Dynamic Panel Data Models with Cross-Sectional Heteroskedasticity

Robust Standard Errors in Transformed Likelihood Estimation of Dynamic Panel Data Models with Cross-Sectional Heteroskedasticity Robust Standard Errors in ransformed Likelihood Estimation of Dynamic Panel Data Models with Cross-Sectional Heteroskedasticity Kazuhiko Hayakawa Hiroshima University M. Hashem Pesaran University of Southern

More information

Short T Dynamic Panel Data Models with Individual and Interactive Time E ects

Short T Dynamic Panel Data Models with Individual and Interactive Time E ects Short T Dynamic Panel Data Models with Individual and Interactive Time E ects Kazuhiko Hayakawa Hiroshima University M. Hashem Pesaran University of Southern California, USA, and Trinity College, Cambridge

More information

1 The Multiple Regression Model: Freeing Up the Classical Assumptions

1 The Multiple Regression Model: Freeing Up the Classical Assumptions 1 The Multiple Regression Model: Freeing Up the Classical Assumptions Some or all of classical assumptions were crucial for many of the derivations of the previous chapters. Derivation of the OLS estimator

More information

Notes on Generalized Method of Moments Estimation

Notes on Generalized Method of Moments Estimation Notes on Generalized Method of Moments Estimation c Bronwyn H. Hall March 1996 (revised February 1999) 1. Introduction These notes are a non-technical introduction to the method of estimation popularized

More information

Subsets Tests in GMM without assuming identi cation

Subsets Tests in GMM without assuming identi cation Subsets Tests in GMM without assuming identi cation Frank Kleibergen Brown University and University of Amsterdam Sophocles Mavroeidis y Brown University sophocles_mavroeidis@brown.edu Preliminary: please

More information

ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Spring 2013 Instructor: Victor Aguirregabiria

ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Spring 2013 Instructor: Victor Aguirregabiria ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Spring 2013 Instructor: Victor Aguirregabiria SOLUTION TO FINAL EXAM Friday, April 12, 2013. From 9:00-12:00 (3 hours) INSTRUCTIONS:

More information

Robust Standard Errors in Transformed Likelihood Estimation of Dynamic Panel Data Models with Cross-Sectional Heteroskedasticity

Robust Standard Errors in Transformed Likelihood Estimation of Dynamic Panel Data Models with Cross-Sectional Heteroskedasticity Robust Standard Errors in ransformed Likelihood Estimation of Dynamic Panel Data Models with Cross-Sectional Heteroskedasticity Kazuhiko Hayakawa Hiroshima University M. Hashem Pesaran USC Dornsife IE

More information

Nonparametric Identi cation and Estimation of Truncated Regression Models with Heteroskedasticity

Nonparametric Identi cation and Estimation of Truncated Regression Models with Heteroskedasticity Nonparametric Identi cation and Estimation of Truncated Regression Models with Heteroskedasticity Songnian Chen a, Xun Lu a, Xianbo Zhou b and Yahong Zhou c a Department of Economics, Hong Kong University

More information

Estimation and Inference with Weak Identi cation

Estimation and Inference with Weak Identi cation Estimation and Inference with Weak Identi cation Donald W. K. Andrews Cowles Foundation Yale University Xu Cheng Department of Economics University of Pennsylvania First Draft: August, 2007 Revised: March

More information

Asymptotic distributions of the quadratic GMM estimator in linear dynamic panel data models

Asymptotic distributions of the quadratic GMM estimator in linear dynamic panel data models Asymptotic distributions of the quadratic GMM estimator in linear dynamic panel data models By Tue Gørgens Chirok Han Sen Xue ANU Working Papers in Economics and Econometrics # 635 May 2016 JEL: C230 ISBN:

More information

ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2016 Instructor: Victor Aguirregabiria

ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2016 Instructor: Victor Aguirregabiria ECOOMETRICS II (ECO 24S) University of Toronto. Department of Economics. Winter 26 Instructor: Victor Aguirregabiria FIAL EAM. Thursday, April 4, 26. From 9:am-2:pm (3 hours) ISTRUCTIOS: - This is a closed-book

More information

GMM estimation of spatial panels

GMM estimation of spatial panels MRA Munich ersonal ReEc Archive GMM estimation of spatial panels Francesco Moscone and Elisa Tosetti Brunel University 7. April 009 Online at http://mpra.ub.uni-muenchen.de/637/ MRA aper No. 637, posted

More information

1 Estimation of Persistent Dynamic Panel Data. Motivation

1 Estimation of Persistent Dynamic Panel Data. Motivation 1 Estimation of Persistent Dynamic Panel Data. Motivation Consider the following Dynamic Panel Data (DPD) model y it = y it 1 ρ + x it β + µ i + v it (1.1) with i = {1, 2,..., N} denoting the individual

More information

LECTURE 12 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT

LECTURE 12 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT MARCH 29, 26 LECTURE 2 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT (Davidson (2), Chapter 4; Phillips Lectures on Unit Roots, Cointegration and Nonstationarity; White (999), Chapter 7) Unit root processes

More information

Comment on HAC Corrections for Strongly Autocorrelated Time Series by Ulrich K. Müller

Comment on HAC Corrections for Strongly Autocorrelated Time Series by Ulrich K. Müller Comment on HAC Corrections for Strongly Autocorrelated ime Series by Ulrich K. Müller Yixiao Sun Department of Economics, UC San Diego May 2, 24 On the Nearly-optimal est Müller applies the theory of optimal

More information

Speci cation of Conditional Expectation Functions

Speci cation of Conditional Expectation Functions Speci cation of Conditional Expectation Functions Econometrics Douglas G. Steigerwald UC Santa Barbara D. Steigerwald (UCSB) Specifying Expectation Functions 1 / 24 Overview Reference: B. Hansen Econometrics

More information

Comparing the asymptotic and empirical (un)conditional distributions of OLS and IV in a linear static simultaneous equation

Comparing the asymptotic and empirical (un)conditional distributions of OLS and IV in a linear static simultaneous equation Comparing the asymptotic and empirical (un)conditional distributions of OLS and IV in a linear static simultaneous equation Jan F. Kiviet and Jerzy Niemczyk y January JEL-classi cation: C3, C, C3 Keywords:

More information

Pseudo panels and repeated cross-sections

Pseudo panels and repeated cross-sections Pseudo panels and repeated cross-sections Marno Verbeek November 12, 2007 Abstract In many countries there is a lack of genuine panel data where speci c individuals or rms are followed over time. However,

More information

Improving GMM efficiency in dynamic models for panel data with mean stationarity

Improving GMM efficiency in dynamic models for panel data with mean stationarity Working Paper Series Department of Economics University of Verona Improving GMM efficiency in dynamic models for panel data with mean stationarity Giorgio Calzolari, Laura Magazzini WP Number: 12 July

More information

Lecture Notes on Measurement Error

Lecture Notes on Measurement Error Steve Pischke Spring 2000 Lecture Notes on Measurement Error These notes summarize a variety of simple results on measurement error which I nd useful. They also provide some references where more complete

More information

Nonparametric Identi cation of a Binary Random Factor in Cross Section Data

Nonparametric Identi cation of a Binary Random Factor in Cross Section Data Nonparametric Identi cation of a Binary Random Factor in Cross Section Data Yingying Dong and Arthur Lewbel California State University Fullerton and Boston College Original January 2009, revised March

More information

11. Bootstrap Methods

11. Bootstrap Methods 11. Bootstrap Methods c A. Colin Cameron & Pravin K. Trivedi 2006 These transparencies were prepared in 20043. They can be used as an adjunct to Chapter 11 of our subsequent book Microeconometrics: Methods

More information

Introduction: structural econometrics. Jean-Marc Robin

Introduction: structural econometrics. Jean-Marc Robin Introduction: structural econometrics Jean-Marc Robin Abstract 1. Descriptive vs structural models 2. Correlation is not causality a. Simultaneity b. Heterogeneity c. Selectivity Descriptive models Consider

More information

The Impact of a Hausman Pretest on the Size of a Hypothesis Test: the Panel Data Case

The Impact of a Hausman Pretest on the Size of a Hypothesis Test: the Panel Data Case The Impact of a Hausman retest on the Size of a Hypothesis Test: the anel Data Case atrik Guggenberger Department of Economics UCLA September 22, 2008 Abstract: The size properties of a two stage test

More information

Chapter 6: Endogeneity and Instrumental Variables (IV) estimator

Chapter 6: Endogeneity and Instrumental Variables (IV) estimator Chapter 6: Endogeneity and Instrumental Variables (IV) estimator Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans December 15, 2013 Christophe Hurlin (University of Orléans)

More information

Economics 241B Estimation with Instruments

Economics 241B Estimation with Instruments Economics 241B Estimation with Instruments Measurement Error Measurement error is de ned as the error resulting from the measurement of a variable. At some level, every variable is measured with error.

More information

x i = 1 yi 2 = 55 with N = 30. Use the above sample information to answer all the following questions. Show explicitly all formulas and calculations.

x i = 1 yi 2 = 55 with N = 30. Use the above sample information to answer all the following questions. Show explicitly all formulas and calculations. Exercises for the course of Econometrics Introduction 1. () A researcher is using data for a sample of 30 observations to investigate the relationship between some dependent variable y i and independent

More information

1 Procedures robust to weak instruments

1 Procedures robust to weak instruments Comment on Weak instrument robust tests in GMM and the new Keynesian Phillips curve By Anna Mikusheva We are witnessing a growing awareness among applied researchers about the possibility of having weak

More information

ECON2285: Mathematical Economics

ECON2285: Mathematical Economics ECON2285: Mathematical Economics Yulei Luo Economics, HKU September 17, 2018 Luo, Y. (Economics, HKU) ME September 17, 2018 1 / 46 Static Optimization and Extreme Values In this topic, we will study goal

More information

We begin by thinking about population relationships.

We begin by thinking about population relationships. Conditional Expectation Function (CEF) We begin by thinking about population relationships. CEF Decomposition Theorem: Given some outcome Y i and some covariates X i there is always a decomposition where

More information

Estimation and Inference with Weak, Semi-strong, and Strong Identi cation

Estimation and Inference with Weak, Semi-strong, and Strong Identi cation Estimation and Inference with Weak, Semi-strong, and Strong Identi cation Donald W. K. Andrews Cowles Foundation Yale University Xu Cheng Department of Economics University of Pennsylvania This Version:

More information

Modeling Expectations with Noncausal Autoregressions

Modeling Expectations with Noncausal Autoregressions Modeling Expectations with Noncausal Autoregressions Markku Lanne * Pentti Saikkonen ** Department of Economics Department of Mathematics and Statistics University of Helsinki University of Helsinki Abstract

More information

The Hausman Test and Weak Instruments y

The Hausman Test and Weak Instruments y The Hausman Test and Weak Instruments y Jinyong Hahn z John C. Ham x Hyungsik Roger Moon { September 7, Abstract We consider the following problem. There is a structural equation of interest that contains

More information

GMM Estimation with Noncausal Instruments

GMM Estimation with Noncausal Instruments ömmföäflsäafaäsflassflassflas ffffffffffffffffffffffffffffffffffff Discussion Papers GMM Estimation with Noncausal Instruments Markku Lanne University of Helsinki, RUESG and HECER and Pentti Saikkonen

More information

MC3: Econometric Theory and Methods. Course Notes 4

MC3: Econometric Theory and Methods. Course Notes 4 University College London Department of Economics M.Sc. in Economics MC3: Econometric Theory and Methods Course Notes 4 Notes on maximum likelihood methods Andrew Chesher 25/0/2005 Course Notes 4, Andrew

More information

On the Power of Tests for Regime Switching

On the Power of Tests for Regime Switching On the Power of Tests for Regime Switching joint work with Drew Carter and Ben Hansen Douglas G. Steigerwald UC Santa Barbara May 2015 D. Steigerwald (UCSB) Regime Switching May 2015 1 / 42 Motivating

More information

Testing Weak Convergence Based on HAR Covariance Matrix Estimators

Testing Weak Convergence Based on HAR Covariance Matrix Estimators Testing Weak Convergence Based on HAR Covariance atrix Estimators Jianning Kong y, Peter C. B. Phillips z, Donggyu Sul x August 4, 207 Abstract The weak convergence tests based on heteroskedasticity autocorrelation

More information

A Conditional-Heteroskedasticity-Robust Con dence Interval for the Autoregressive Parameter

A Conditional-Heteroskedasticity-Robust Con dence Interval for the Autoregressive Parameter A Conditional-Heteroskedasticity-Robust Con dence Interval for the Autoregressive Parameter Donald W. K. Andrews Cowles Foundation for Research in Economics Yale University Patrik Guggenberger Department

More information

A CONDITIONAL-HETEROSKEDASTICITY-ROBUST CONFIDENCE INTERVAL FOR THE AUTOREGRESSIVE PARAMETER. Donald W.K. Andrews and Patrik Guggenberger

A CONDITIONAL-HETEROSKEDASTICITY-ROBUST CONFIDENCE INTERVAL FOR THE AUTOREGRESSIVE PARAMETER. Donald W.K. Andrews and Patrik Guggenberger A CONDITIONAL-HETEROSKEDASTICITY-ROBUST CONFIDENCE INTERVAL FOR THE AUTOREGRESSIVE PARAMETER By Donald W.K. Andrews and Patrik Guggenberger August 2011 Revised December 2012 COWLES FOUNDATION DISCUSSION

More information

1. GENERAL DESCRIPTION

1. GENERAL DESCRIPTION Econometrics II SYLLABUS Dr. Seung Chan Ahn Sogang University Spring 2003 1. GENERAL DESCRIPTION This course presumes that students have completed Econometrics I or equivalent. This course is designed

More information

Robust Con dence Intervals in Nonlinear Regression under Weak Identi cation

Robust Con dence Intervals in Nonlinear Regression under Weak Identi cation Robust Con dence Intervals in Nonlinear Regression under Weak Identi cation Xu Cheng y Department of Economics Yale University First Draft: August, 27 This Version: December 28 Abstract In this paper,

More information

UNIVERSITY OF CALIFORNIA, SAN DIEGO DEPARTMENT OF ECONOMICS

UNIVERSITY OF CALIFORNIA, SAN DIEGO DEPARTMENT OF ECONOMICS 2-7 UNIVERSITY OF LIFORNI, SN DIEGO DEPRTMENT OF EONOMIS THE JOHNSEN-GRNGER REPRESENTTION THEOREM: N EXPLIIT EXPRESSION FOR I() PROESSES Y PETER REINHRD HNSEN DISUSSION PPER 2-7 JULY 2 The Johansen-Granger

More information

ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2014 Instructor: Victor Aguirregabiria

ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2014 Instructor: Victor Aguirregabiria ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2014 Instructor: Victor guirregabiria SOLUTION TO FINL EXM Monday, pril 14, 2014. From 9:00am-12:00pm (3 hours) INSTRUCTIONS:

More information

Tests for Cointegration, Cobreaking and Cotrending in a System of Trending Variables

Tests for Cointegration, Cobreaking and Cotrending in a System of Trending Variables Tests for Cointegration, Cobreaking and Cotrending in a System of Trending Variables Josep Lluís Carrion-i-Silvestre University of Barcelona Dukpa Kim y Korea University May 4, 28 Abstract We consider

More information

Bias Correction Methods for Dynamic Panel Data Models with Fixed Effects

Bias Correction Methods for Dynamic Panel Data Models with Fixed Effects MPRA Munich Personal RePEc Archive Bias Correction Methods for Dynamic Panel Data Models with Fixed Effects Mohamed R. Abonazel Department of Applied Statistics and Econometrics, Institute of Statistical

More information

Specification Test for Instrumental Variables Regression with Many Instruments

Specification Test for Instrumental Variables Regression with Many Instruments Specification Test for Instrumental Variables Regression with Many Instruments Yoonseok Lee and Ryo Okui April 009 Preliminary; comments are welcome Abstract This paper considers specification testing

More information

Dynamic Panels. Chapter Introduction Autoregressive Model

Dynamic Panels. Chapter Introduction Autoregressive Model Chapter 11 Dynamic Panels This chapter covers the econometrics methods to estimate dynamic panel data models, and presents examples in Stata to illustrate the use of these procedures. The topics in this

More information

Lecture 4: Linear panel models

Lecture 4: Linear panel models Lecture 4: Linear panel models Luc Behaghel PSE February 2009 Luc Behaghel (PSE) Lecture 4 February 2009 1 / 47 Introduction Panel = repeated observations of the same individuals (e.g., rms, workers, countries)

More information

Economics 241B Review of Limit Theorems for Sequences of Random Variables

Economics 241B Review of Limit Theorems for Sequences of Random Variables Economics 241B Review of Limit Theorems for Sequences of Random Variables Convergence in Distribution The previous de nitions of convergence focus on the outcome sequences of a random variable. Convergence

More information

Panel VAR Models with Spatial Dependence

Panel VAR Models with Spatial Dependence Panel VAR Models with Spatial Dependence Jan Mutl Institute of Advanced Studies Stumpergasse 56 A-6 Vienna Austria mutl@ihs.ac.at January 3, 29 Abstract I consider a panel vector autoregressive (panel

More information

Weak - Convergence: Theory and Applications

Weak - Convergence: Theory and Applications Weak - Convergence: Theory and Applications Jianning Kong y, Peter C. B. Phillips z, Donggyu Sul x October 26, 2018 Abstract The concept of relative convergence, which requires the ratio of two time series

More information

Equivalence of several methods for decomposing time series into permananent and transitory components

Equivalence of several methods for decomposing time series into permananent and transitory components Equivalence of several methods for decomposing time series into permananent and transitory components Don Harding Department of Economics and Finance LaTrobe University, Bundoora Victoria 3086 and Centre

More information

An estimate of the long-run covariance matrix, Ω, is necessary to calculate asymptotic

An estimate of the long-run covariance matrix, Ω, is necessary to calculate asymptotic Chapter 6 ESTIMATION OF THE LONG-RUN COVARIANCE MATRIX An estimate of the long-run covariance matrix, Ω, is necessary to calculate asymptotic standard errors for the OLS and linear IV estimators presented

More information

1 Introduction The time series properties of economic series (orders of integration and cointegration) are often of considerable interest. In micro pa

1 Introduction The time series properties of economic series (orders of integration and cointegration) are often of considerable interest. In micro pa Unit Roots and Identification in Autoregressive Panel Data Models: A Comparison of Alternative Tests Stephen Bond Institute for Fiscal Studies and Nuffield College, Oxford Céline Nauges LEERNA-INRA Toulouse

More information

CAE Working Paper # Fixed-b Asymptotic Approximation of the Sampling Behavior of Nonparametric Spectral Density Estimators

CAE Working Paper # Fixed-b Asymptotic Approximation of the Sampling Behavior of Nonparametric Spectral Density Estimators CAE Working Paper #06-04 Fixed-b Asymptotic Approximation of the Sampling Behavior of Nonparametric Spectral Density Estimators by Nigar Hashimzade and Timothy Vogelsang January 2006. Fixed-b Asymptotic

More information

GROWTH AND CONVERGENCE IN A MULTI-COUNTRY EMPIRICAL STOCHASTIC SOLOW MODEL

GROWTH AND CONVERGENCE IN A MULTI-COUNTRY EMPIRICAL STOCHASTIC SOLOW MODEL JOURAL OF APPLIED ECOOMETRICS, VOL. 12, 357±392 (1997) GROWTH AD COVERGECE I A MULTI-COUTRY EMPIRICAL STOCHASTIC SOLOW MODEL KEVI LEE, a M. HASHEM PESARA b AD RO SMITH c a Department of Economics, University

More information

Modeling Expectations with Noncausal Autoregressions

Modeling Expectations with Noncausal Autoregressions MPRA Munich Personal RePEc Archive Modeling Expectations with Noncausal Autoregressions Markku Lanne and Pentti Saikkonen Department of Economics, University of Helsinki, Department of Mathematics and

More information

Models, Testing, and Correction of Heteroskedasticity. James L. Powell Department of Economics University of California, Berkeley

Models, Testing, and Correction of Heteroskedasticity. James L. Powell Department of Economics University of California, Berkeley Models, Testing, and Correction of Heteroskedasticity James L. Powell Department of Economics University of California, Berkeley Aitken s GLS and Weighted LS The Generalized Classical Regression Model

More information

Bootstrapping Long Memory Tests: Some Monte Carlo Results

Bootstrapping Long Memory Tests: Some Monte Carlo Results Bootstrapping Long Memory Tests: Some Monte Carlo Results Anthony Murphy and Marwan Izzeldin Nu eld College, Oxford and Lancaster University. December 2005 - Preliminary Abstract We investigate the bootstrapped

More information

Some Recent Developments in Spatial Panel Data Models

Some Recent Developments in Spatial Panel Data Models Some Recent Developments in Spatial Panel Data Models Lung-fei Lee Department of Economics Ohio State University l ee@econ.ohio-state.edu Jihai Yu Department of Economics University of Kentucky jihai.yu@uky.edu

More information

GMM Based Tests for Locally Misspeci ed Models

GMM Based Tests for Locally Misspeci ed Models GMM Based Tests for Locally Misspeci ed Models Anil K. Bera Department of Economics University of Illinois, USA Walter Sosa Escudero University of San Andr es and National University of La Plata, Argentina

More information

Bootstrapping Long Memory Tests: Some Monte Carlo Results

Bootstrapping Long Memory Tests: Some Monte Carlo Results Bootstrapping Long Memory Tests: Some Monte Carlo Results Anthony Murphy and Marwan Izzeldin University College Dublin and Cass Business School. July 2004 - Preliminary Abstract We investigate the bootstrapped

More information

The Predictive Space or If x predicts y; what does y tell us about x?

The Predictive Space or If x predicts y; what does y tell us about x? The Predictive Space or If x predicts y; what does y tell us about x? Donald Robertson and Stephen Wright y October 28, 202 Abstract A predictive regression for y t and a time series representation of

More information

SIMILAR-ON-THE-BOUNDARY TESTS FOR MOMENT INEQUALITIES EXIST, BUT HAVE POOR POWER. Donald W. K. Andrews. August 2011

SIMILAR-ON-THE-BOUNDARY TESTS FOR MOMENT INEQUALITIES EXIST, BUT HAVE POOR POWER. Donald W. K. Andrews. August 2011 SIMILAR-ON-THE-BOUNDARY TESTS FOR MOMENT INEQUALITIES EXIST, BUT HAVE POOR POWER By Donald W. K. Andrews August 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1815 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS

More information

Combining Macroeconomic Models for Prediction

Combining Macroeconomic Models for Prediction Combining Macroeconomic Models for Prediction John Geweke University of Technology Sydney 15th Australasian Macro Workshop April 8, 2010 Outline 1 Optimal prediction pools 2 Models and data 3 Optimal pools

More information

Testing for a Trend with Persistent Errors

Testing for a Trend with Persistent Errors Testing for a Trend with Persistent Errors Graham Elliott UCSD August 2017 Abstract We develop new tests for the coe cient on a time trend in a regression of a variable on a constant and time trend where

More information

Chapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE

Chapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE Chapter 6. Panel Data Joan Llull Quantitative Statistical Methods II Barcelona GSE Introduction Chapter 6. Panel Data 2 Panel data The term panel data refers to data sets with repeated observations over

More information

Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models

Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models Fall 22 Contents Introduction 2. An illustrative example........................... 2.2 Discussion...................................

More information

Spatial panels: random components vs. xed e ects

Spatial panels: random components vs. xed e ects Spatial panels: random components vs. xed e ects Lung-fei Lee Department of Economics Ohio State University l eeecon.ohio-state.edu Jihai Yu Department of Economics University of Kentucky jihai.yuuky.edu

More information

Efficient Estimation of Dynamic Panel Data Models: Alternative Assumptions and Simplified Estimation

Efficient Estimation of Dynamic Panel Data Models: Alternative Assumptions and Simplified Estimation Efficient Estimation of Dynamic Panel Data Models: Alternative Assumptions and Simplified Estimation Seung C. Ahn Arizona State University, Tempe, AZ 85187, USA Peter Schmidt * Michigan State University,

More information

ECONOMETRICS II (ECO 2401) Victor Aguirregabiria (March 2017) TOPIC 1: LINEAR PANEL DATA MODELS

ECONOMETRICS II (ECO 2401) Victor Aguirregabiria (March 2017) TOPIC 1: LINEAR PANEL DATA MODELS ECONOMETRICS II (ECO 2401) Victor Aguirregabiria (March 2017) TOPIC 1: LINEAR PANEL DATA MODELS 1. Panel Data 2. Static Panel Data Models 2.1. Assumptions 2.2. "Fixed e ects" and "Random e ects" 2.3. Estimation

More information

Rank Estimation of Partially Linear Index Models

Rank Estimation of Partially Linear Index Models Rank Estimation of Partially Linear Index Models Jason Abrevaya University of Texas at Austin Youngki Shin University of Western Ontario October 2008 Preliminary Do not distribute Abstract We consider

More information

CAE Working Paper # A New Asymptotic Theory for Heteroskedasticity-Autocorrelation Robust Tests. Nicholas M. Kiefer and Timothy J.

CAE Working Paper # A New Asymptotic Theory for Heteroskedasticity-Autocorrelation Robust Tests. Nicholas M. Kiefer and Timothy J. CAE Working Paper #05-08 A New Asymptotic Theory for Heteroskedasticity-Autocorrelation Robust Tests by Nicholas M. Kiefer and Timothy J. Vogelsang January 2005. A New Asymptotic Theory for Heteroskedasticity-Autocorrelation

More information

Economics 620, Lecture 14: Lags and Dynamics

Economics 620, Lecture 14: Lags and Dynamics Economics 620, Lecture 14: Lags and Dynamics Nicholas M. Kiefer Cornell University Professor N. M. Kiefer (Cornell University) Lecture 14: Lags and Dynamics 1 / 15 Modern time-series analysis does not

More information

A New Computational Algorithm for Random Coe cients Model with Aggregate-level Data

A New Computational Algorithm for Random Coe cients Model with Aggregate-level Data A New Computational Algorithm for Random Coe cients Model with Aggregate-level Data Jinyoung Lee y UCLA February 9, 011 Abstract The norm for estimating di erentiated product demand with aggregate-level

More information

Nearly Unbiased Estimation in Dynamic Panel Data Models

Nearly Unbiased Estimation in Dynamic Panel Data Models TI 00-008/ Tinbergen Institute Discussion Paper Nearly Unbiased Estimation in Dynamic Panel Data Models Martin A. Carree Department of General Economics, Faculty of Economics, Erasmus University Rotterdam,

More information

The Time Series and Cross-Section Asymptotics of Empirical Likelihood Estimator in Dynamic Panel Data Models

The Time Series and Cross-Section Asymptotics of Empirical Likelihood Estimator in Dynamic Panel Data Models The Time Series and Cross-Section Asymptotics of Empirical Likelihood Estimator in Dynamic Panel Data Models Günce Eryürük August 200 Centro de Investigación Económica, Instituto Tecnológico Autónomo de

More information